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Astron. Astrophys. 359, 1085-1106 (2000)

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3. Discussion

In this section we evaluate the advantages of time consuming non-LTE calculations over a straightforward LTE analysis for the interpretation of the OI spectra of BA-type stars. First, we present the results of our model calculations in the form of a grid of equivalent widths and non-LTE corrections for a set of diagnostic OI lines applicable to abundance studies. In the following the nature of the non-LTE effects is discussed for the parameter space of BA-type stars. The problems with modelling the observed strong near-infrared lines are addressed and finally our results are compared with those of previous studies of non-LTE effects in atomic oxygen.

3.1. Results of our model calculations

Fig. 5 summarises our results on the OI lines [FORMULA] 3947, 5329-30, 6155-8, 7002 and 7771-5. Displayed are the computed equivalent widths [FORMULA] from LTE and non-LTE calculations summed over all components of a transition for a given oxygen abundance

[EQUATION]

and expected non-LTE abundance corrections

[EQUATION]

as a function of [FORMULA] for surface gravities typical for main sequence stars, Ib and Ia supergiants. The [FORMULA] are calculated from LTE abundances adjusted to reproduce the equivalent widths obtained in the non-LTE computation. In order to account for effects introduced by the variation of other important stellar parameters three sets of calculations were performed, for microturbulences of [FORMULA]=2 and 8 km s-1 at solar elemental abundances and for [FORMULA]=2 km s-1 at a composition reduced by 0.7 dex as expected for e.g. SMC objects. The oxygen abundance is kept fixed at the solar value in the latter case. This small inconsistency is acceptable as we are mainly interested in the effects of an increased UV ionization flux on the oxygen ionization balance. A reduced oxygen content will confine the line formation to deeper layers.

[FIGURE] Fig. 5. Theoretical equivalent widths [FORMULA] (in mÅ) and non-LTE corrections [FORMULA] (as defined by Eq. (3)) as a function of [FORMULA] for values of [FORMULA] suitable for stars of luminosity class V, Ib and Ia. Solid line: [FORMULA], [FORMULA], [FORMULA]; dotted line: [FORMULA], [FORMULA], [FORMULA]; dashed line: [FORMULA], [FORMULA], [FORMULA]. The non-LTE equivalent widths are larger than their LTE counterparts. Note the different temperature range for the Ia results. For [FORMULA] 6155-8 observed [FORMULA] according to Takeda & Takada-Hidai (1998) are also displayed. Triangles, squares, diamonds: Ib, Iab, Ia supergiants.

[FIGURE] Fig. 5. (continued) For the near-infrared triplet [FORMULA] 7771-5 observed [FORMULA] according to Faraggiana et al. (1988) are also displayed. Circles: main sequence stars; crosses: subgiants; triangles, squares, diamonds: Ib, Iab, Ia supergiants.

Several important conclusions can be drawn from Fig. 5. As expected, the predicted equivalent widths for the BA-type stars decrease monotonically with increasing temperature as oxygen becomes more and more ionized (cf. Fig. 6); at constant [FORMULA] the ionization balance is shifted to the higher ionization stage at lower surface gravities but this is overcompensated by an increased population of the excited energy levels, resulting in a strengthening of the lines. Electron collisions become less effective in the low density conditions in supergiants thus favouring departures from LTE. The non-LTE abundance corrections also increase with temperature as deviations from LTE occur in an increasing fraction of the line formation region. In the late A-types only the line centres are affected by non-LTE (see also Sect. 3.1) resulting in small non-LTE corrections.

[FIGURE] Fig. 6. Ratio of OI to OII non-LTE populations as a function of [FORMULA] for different stellar parameters [FORMULA]. The same line identifiers as in Fig. 5 are used. For the [FORMULA] model at solar oxygen content and metallicity the LTE ratio is also displayed (thick dots).

Non-LTE effects on the weak OI lines in the visible are negligible for main sequence stars but the situation changes markedly in supergiants. For objects close to the Eddington limit non-LTE corrections can amount to [FORMULA]0.5 dex even for these weak lines. Observed equivalent widths from Takeda & Takada-Hidai (1998) are included in Fig. 5 for OI [FORMULA] 6155-8 with their effective temperatures adopted. Note that the gravities of these stars span the range [FORMULA] thus introducing some of the scatter around our predictions for fixed [FORMULA]. Moreover, Takeda & Takada-Hidai (1998) also find a reduced iron abundance for most of those stars with significantly smaller [FORMULA]6155-8) thus implying a general metal underabundance for these objects.

The near-infrared triplet on the other hand is significantly affected by non-LTE effects even on the main sequence, reaching a dramatic non-LTE strengthening in supergiants. In the case of [FORMULA] 7771-5 observed equivalent widths from Faraggiana et al. (1988) are also displayed in Fig. 5. Effective temperatures are assigned to spectral types via the relations given by Gray (1992) and Humphreys & McElroy (1984) for main sequence stars and for supergiants, respectively. Good agreement between theory and observation is found in a statistical sense near the main sequence despite the fact that the stellar parameters - and individual oxygen abundances - may differ from our assumed values. For supergiants the discrepancies between theory and observation increase with increasing luminosity. This trend is not due to small number statistics but is a genuine effect as will be shown later.

Microturbulence has no significant influence on the non-LTE abundance corrections apart from its classical effect of strengthening lines on the flat part of the curve of growth. Our (moderate) changes in metallicity also have no strong impact on the [FORMULA].

The weak lines should be viewed as the spectral features reproduced most convincingly by theoretical means and therefore are to be preferred in abundance analyses. A much greater challenge is posed by the modelling of the observed strong near-infrared lines [FORMULA] 7771-5 and 8446 in supergiants for which we cannot obtain results consistent with those from the observed weak lines in our simple approach. The discussion of these problems will be resumed later, but we first wish to gain some quantitative insight into the dependence of the non-LTE effects on the model parameters.

3.2. The non-LTE effects

The non-LTE ionization balance for oxygen at various stellar parameters is displayed in Fig. 6. At mid-A type temperatures OI is the dominant ionization stage but oxygen rapidly ionizes with increasing [FORMULA] and decreasing [FORMULA]. The variations of the ionization balance with microturbulence and metallicity reflect the changes in the line blanketing at these parameters. In addition, the non-LTE ionization balance for oxygen deviates only marginally from that under LTE conditions as displayed in Fig. 6 for selected cases.

Departure coefficients [FORMULA] for the energy levels i are displayed in Fig. 7 as a function of the Rosseland optical depth [FORMULA] for main sequence and supergiant models of mid-/early A-type and late B stars. Furthermore, the [FORMULA] for an object close to the Eddington limit are displayed in Fig. 8 with additional energy terms. All the observed lines originate from the energy levels [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

[FIGURE] Fig. 7. Departure coefficients [FORMULA] of term i as a function of Rosseland optical depth [FORMULA]. Solid line: [FORMULA], [FORMULA], [FORMULA]; dotted line: [FORMULA], [FORMULA], [FORMULA]; dashed line: [FORMULA], [FORMULA], [FORMULA]. The formation depths of the line core ([FORMULA]) for several transitions are indicated. Term identifiers - 1: [FORMULA]; 4: [FORMULA]; 5: [FORMULA]; 6: [FORMULA]; 7: [FORMULA]; 52: [FORMULA] (OII )

[FIGURE] Fig. 8. Departure coefficients [FORMULA] as a function of [FORMULA] from the model calculation for the supergiant HD 92207. In addition to those from Fig. 7 the [FORMULA] from the following energy terms are also displayed - 2: [FORMULA]; 3: [FORMULA]; 12: [FORMULA]; 13: [FORMULA]; 15: [FORMULA]; 17: [FORMULA]; 18: [FORMULA]; 25: [FORMULA]. The formation depths of the line core ([FORMULA]) for the main diagnostic lines are indicated.

Deep in the atmosphere the departure coefficients approach unity as the density increases and collisional processes dominate, enforcing LTE (inner boundary condition). Farther out in the atmosphere marked deviations from LTE occupations occur, setting in at larger [FORMULA] and being more pronounced in supergiants. Strong overpopulations are found for the metastable level [FORMULA] and for [FORMULA], the lower states for the [FORMULA] 7771-5 and 8446 transitions, respectively. Overpopulations by a factor of [FORMULA]4 occur in main sequence stars and 6-16 in supergiants at maximum. This behaviour may be understood in terms of recombinations cascading to lower n via transitions among the levels with quantum numbers (n, [FORMULA]). The radiative downward rates feed the [FORMULA] level populations both in the triplet and the quintet spin system. Electron collisions are ineffective in depopulating the metastable [FORMULA] state and the [FORMULA] level. Moreover, the latter energy level also gains metastable character as the net radiative rate to the OI ground state turns out to be essentially zero. However, the two [FORMULA] levels are close energetically and can be coupled collisionally at higher densities despite the small collision strength of this (octupole) transition. Thus their departure coefficients behave similarly. The other level populations show much smaller deviations from LTE, less than a factor of 2 even in the supergiants. The [FORMULA] levels - the lower levels of the observed weak OI lines - show a moderate overpopulation due to being part of the recombination cascade mentioned above and the strong radiative coupling with the [FORMULA] levels. Collisions couple these two levels in the same manner as for the [FORMULA] levels at a larger collision strength. In general, accounting for the detailed collision cross sections is essential to determine the non-LTE corrections quantitatively as the lines in the visible and IR are strongly influenced by the collisional processes even in supergiants. Finally, the ground states of OI /OII and the first two excited singlet levels coupled with the OI ground state via strong collision rates deviate only marginally from LTE. This has to be checked for non-LTE atmospheric models where the hydrogen non-LTE departures (negligible here) are expected to be forced upon oxygen via charge exchange. The qualitative behaviour of the [FORMULA] for the HD 92207 model is similar to the (10000/1.5) calculation with non-LTE effects setting in deeper in the atmosphere and the departures being more pronounced.

We have just discussed the importance of recombination cascades for the strengthening of the observed oxygen lines. A problem in this context might arise from insufficient collisional coupling of the highest energy levels treated explicitly in our statistical equilibrium calculations with the continuum. As a test we increased the coupling of these levels to the continuum by a factor of [FORMULA]. The populations of the lower levels of the observed lines change by less than 2% with negligible effects on the calculated equivalent widths.

The formation depths of the line core at [FORMULA] for the strongest lines in the near-infrared and the visible are also indicated in Fig. 7. All other weak lines in the visible are formed even deeper in the atmosphere than [FORMULA] 6158 (cf. also Fig. 8). The extent of the non-LTE abundance corrections presented in Fig. 5 can be qualitatively deduced from the [FORMULA]-diagrams and the behaviour of the line source function [FORMULA] as shown exemplary in Fig. 9. A reduction of the metal content in the atmospheric model and the microturbulence affect the departure coefficients only moderately.

[FIGURE] Fig. 9. Ratio of line source function [FORMULA] to Planck function [FORMULA] at line centre as a function of [FORMULA] for the supergiant HD 92207. The formation depths of the line cores ([FORMULA]) are indicated

The ratio of the line source function to the Planck function

[EQUATION]

for an early A-type supergiant model (HD 92207, see Sect. 4) is displayed in Fig. 9, h being the Planck constant, [FORMULA] the transition frequency, k the Boltzmann constant, T the temperature, n the occupation numbers and g the statistical weights of the lower/upper level i/j. The behaviour of [FORMULA] is qualitatively the same for the supergiant models in our parameter range. Deviations of [FORMULA] from [FORMULA] set in deeper in the atmosphere with increasing [FORMULA] and decreasing [FORMULA]. For the near-infrared lines a marked reduction of the line centre intensity is expected due to photon escape (see e.g. Mihalas (1978), Ch. 11-2). This also affects the lines in the visible. Again, variations of microturbulence and metallicity result in only moderate changes as tests have shown. In the main sequence models the non-LTE effects on the line source function are dramatically reduced, deviating from [FORMULA]=1 only in the outer formation region of the strong near-infrared lines.

A simple approximation to the behaviour of the source functions of the strong near-infrared lines is given by the classical two-level atom. The [FORMULA] from the detailed calculation and the simple model resemble each other throughout the line formation region.

Additionally, the ratio of non-LTE to LTE line opacities

[EQUATION]

is found to mainly follow the departure coefficients of the lower levels of the transitions. Thus especially for the strong near-infrared lines a marked increase in the line opacity as compared to LTE is expected. The lines are strengthened enormously due to the (pseudo-)metastability of the lower levels.

3.3. The strong near-infrared lines

Fig. 7 also offers some indication for the reasons for our failure to reproduce the observed strong near-infrared lines. In contrast to the weak lines they are tracers for the physical structure of the stellar atmosphere over a considerable part of its geometrical extent. However, at optical depths [FORMULA] non-LTE effects on the model structure of Ib supergiants become more and more important as found by Przybilla (1997). This is the case at even larger [FORMULA] in the atmospheres of the more luminous supergiants. Furthermore, spherical extension of the atmosphere and outflow velocity fields present at the base of the stellar wind of the supergiants will alter the conditions for the line formation significantly. A reinvestigation will shed light on these points as improved model atmospheres become available.

Weaknesses in our OI model atom (especially in the collision rates) might also be present. These may be investigated most easily in main sequence stars where the atmospheric structure is sufficiently well described by the standard assumptions of being plane-parallel, homogeneous, stationary and of being in hydrostatic and radiative equilibrium. In Fig. 10 the results from model calculations with two sets of collisional data are compared with the observed near-infrared lines of Vega. Obviously a significant improvement can be achieved by avoiding widely used approximation formulae for the collisional processes. Therefore the collisional data of Bhatia & Kastner (1995) is used for the rest of our work. Unfortunately, detailed data on collisions are scarce and there is no possibility at present to check whether the remaining differences between theory and observation result from this deficiency or whether alternative explanations can be found (as discussed below).

[FIGURE] Fig. 10. The importance of collisional cross sections: theoretical line profiles from different atomic models for [FORMULA]=8.59 (as derived in Sect. 4.2) are compared with those observed for Vega (full line). Dotted: collision strengths from Bhatia & Kastner (1995) for most of the transitions between energy levels with [FORMULA] adopted; dashed: Van Regemorter (1962) and Allen (1973) approximation ([FORMULA]=1 for the latter) for these transitions assumed.

In anticipation of the spectra analysed in the next section, we discuss our theoretical results for [FORMULA] 7771-5 in comparison with the observations for the main sequence star Vega and the two supergiants [FORMULA] Leo and HD 92207 in order to show what can be achieved within our approach. The observed profiles for this extremely strong triplet ([FORMULA] larger than those of the strongest Balmer lines in supergiants) are displayed in Fig. 11 together with results from our computations in non-LTE and LTE, see the figure caption for details.

[FIGURE] Fig. 11. Comparison of theoretical line profiles for OI [FORMULA] 7771-5 with those observed for Vega, [FORMULA] Leo and HD 92207 (full line). Dotted: non-LTE line-formation for [FORMULA] = 8.87/8.92/9.85 (Vega/[FORMULA] Leo/ HD 92207); dashed: LTE line-formation for the same abundances; dashed-dotted: LTE line-formation for [FORMULA] = 9.20/10.38/10.96 ([FORMULA] of the observed lines reproduced), cf. Table 3.

The observed profiles for [FORMULA] 7771-5 cannot be exactly reproduced with the abundances determined from the weak lines, neither in non-LTE nor in LTE. They indicate a higher oxygen abundance. The absorption coefficient at line centre is increased in non-LTE in addition to a depression of the line source function due to photon escape, resulting in much deeper line profiles compared to the LTE profiles for the same oxygen abundance. Only in the wings do they approach the values given by LTE. But the equivalent widths derived for both are too small compared to those observed as the computed profiles are not broad enough in the case of supergiants. A further LTE computation is also displayed that matches the observed [FORMULA]. In this case the line centre is not deep enough but marked damping wings begin to develop, especially in the supergiants. These are not present in the observations. The situation is similar in OI [FORMULA] 8446. We conclude that equivalent-width studies of the strong near-infrared OIlines are inadequate for abundance determinations . While non-LTE line formation for these lines occurs on the flat part of the curve of growth, in LTE the large abundances required imply that the formation takes place on the damping part.

For an accurate representation of the line profiles a number of parameters have to be correctly determined. Varying the projected rotational velocity [FORMULA] and the (depth independent ) micro- and macroturbulent velocities ([FORMULA], [FORMULA]) for the objects within the range given by Table 2 does not improve the fit significantly. Only changes in the microturbulence parameter alter the equivalent width, variation of the other parameters only results in profile changes. The limits are set by numerous other metallic lines throughout the spectra. Furthermore, our Stark broadening parameters for the lines might be inaccurate but tight limits are set for the case of Vega and consequently the values needed to fit the supergiant line wings can be excluded. Weaknesses in our model atom/atmospheres might be another possibility but no definitive conclusions can be drawn as e.g. a depth dependent microturbulence claimed by other authors will resolve the problem, see Sect. 5 for details.


[TABLE]

Table 2. Basic properties and atmospheric parameters for the test stars



[TABLE]

Table 3. Abundance analysis for oxygen in the test stars.
a) second entry for a depth-dependent microturbulence, see Sect. 5;
b) omitting results displayed in italics; the mean and the standard deviation in the column of [FORMULA] give the corresponding LTE values



[TABLE]

Table 4. Accuracy of gf values


3.4. Comparison with other studies

Several other studies on non-LTE effects for neutral oxygen have been carried out in the past. Two of them deal with the general problem rather than concentrate on particular details so that a comparison with our results is desirable.

Baschek et al. (1977) discuss an OI model consisting of the ground state and the first seven excited energy levels of the quintet spin system plus the continuum. The non-LTE abundance corrections they find are systematically larger than ours. Also the maximum in the line strengths occurs around 10000 K in their calculations in contrast to our results and the available observations (cf. Fig. 5). A closer inspection of their departure coefficients for stellar parameters comparable with ours shows qualitatively similar behaviour; but the non-LTE departures set in deeper in the atmosphere and are also more pronounced. We suppose that their neglect of metal line-blanketing is an important factor in this context as we also find a strengthening of the non-LTE effects when the metallic background opacities are omitted. Note that the modification of the background opacities also results in a change of the atmospheric structure. Due to the absent backwarming effect their models show a reduced local temperature at the line formation region thus explaining the shift of line strength maximum at least qualitatively. As their model atom is quite limited in the energy levels considered and the atomic data also somewhat outdated final conclusions for the discrepancy with our results cannot be drawn. We cannot implement their model atom in DETAIL/SURFACE for further comparison as they used unpublished photoionization cross-sections.

In their OI analysis for Vega they do not find consistent abundances from different lines but they note that the equivalent widths used in this process might be influenced by unidentified line blends in the moderate resolution spectra dating back to Matsushima & Groth (1960).

A comprehensive non-LTE model for neutral oxygen is presented by Takeda (1992). Again, the derived non-LTE abundance corrections are larger than ours as are the departure coefficients [FORMULA]. For some levels the behaviour is not even qualitatively similar to that in our work. Tracing the discrepancies back to their origins is difficult as insufficient details are provided. Line-blanketing is accounted for by older ODFs (Kurucz 1979), therefore the background opacities should be somewhat smaller, resulting in strengthened non-LTE departures. In general, the atomic data and even the gf values differ. For Vega abundances from the strong OI lines are slightly underestimated in this model (Takeda 1993) when compared to the weak line results, contrary to our findings. This trend increases dramatically in the calculations for the supergiant [FORMULA] Cyg (Takeda 1992). Whether this is due to an overestimation of the non-LTE effects or due to inaccurate stellar parameters cannot be decided here. Furthermore, as above, line blends might be unaccounted for in the equivalent-width determination.

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Online publication: July 13, 2000
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